Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-4x-3y &= 1 \\ -5x-2y &= -3\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-2$ and the bottom equation by $3$ $\begin{align*}8x+6y &= -2\\ -15x-6y &= -9\end{align*}$ Add the top and bottom equations. $-7x = -11$ Divide both sides by $-7$ and reduce as necessary. $x = \dfrac{11}{7}$ Substitute $\dfrac{11}{7}$ for $x$ in the top equation. $-4( \dfrac{11}{7})-3y = 1$ $-\dfrac{44}{7}-3y = 1$ $-3y = \dfrac{51}{7}$ $y = -\dfrac{17}{7}$ The solution is $\enspace x = \dfrac{11}{7}, \enspace y = -\dfrac{17}{7}$.